Neutrino Oscillations

Suggested reading:

C. Giunti and C.W. Kim, Fundamentals of Neutrino Physics and Astrophysics, Oxford University Press (2007; 728 pages)

Elisa BernardiniDeutsches Elektronen-Synchrotron DESY (Zeuthen)

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Neutrino oscillations in vacuum • Idea: interference of different massive neutrinos

• The mass differences must be small: ν’s are produced and

detected coherently• Assumptions for the derivation of oscillation probability:

– Neutrinos are ultra-relativistic– Neutrinos are produced in a defined flavor state

– The experimental resolution in energy-momentum does not allow the determination of the individual masses

• Neutrino flavor states να can then be described by

a superposition of mass (Hamiltonian) eigenstates νk: Eq. 1

Bruno Pontecorvo

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Neutrino propagation• The Schroedinger equation implies that they evolve in

time as plane waves:

• The time evolution of a flavor state is then:

• U†U=1 implies that:

• Combining Eq 2 and Eq 3 we obtain that a pure flavor state at t=0 becomes a superposition of different flavors states at t>0

Eq. 3

Eq. 2

Eq. 4

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Neutrino oscillation probability• The coefficient

gives the probability of transition as a function of time:

• For relativistic neutrinos:

Eq. 5

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Neutrino oscillations in vacuum• The transition probability between two different states is then

• In experiments, the distance to the source L is measured (not the time t)

• Neutrino oscillations can shed light on the squared-mass differences and the elements of the mixing matrix

• Phases of neutrino oscillations:

• Symmetry transformation of states:– CP violated:– T violated:– CPT conserved:

Eq. 6

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Two-neutrino mixing• Consider only two massive neutrinos out of three• The two flavor states are superposition of the two mass

states ν1 and ν2 with coefficients given by the elements of the effective mixing matrix:

• There is one mixing angle 0 ≤ θ ≤ π/2 and one squared-mass difference Δm2

• The transition probability (α≠β) is:

• And its average (in energy and distances):

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• Two classes of experiments:– Appearance: observe

transitions between different flavors

– Disappearance: measure the survival probability of a flavor

• According to the ratio L/E:– The transition between

flavors cannot be observed if

– Only the average of the transition probability manifests itself if

Sensitivity to neutrino oscillations

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Strumia, Vissani

Sources of neutrinos

• Reactor/Accelerator• Supernova • Solar• Atmospheric

Neutrino fluxes

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Reactor neutrinos• Fission reactors are strong sources of anti-νe from β-

decay of neutron-rich nuclei (235U, 238U, 239Pu, 241Pu)

• Very intense ~ 1020 s-1 per GWth of thermal power

• Neutrino flux is isotropic

• Energy ~ few MeV (only νe disappearance)

• Anti-νe detected via inverse β-decay (Eth 1.8 MeV)

• Sensitivity to oscillations:– Source-detector distance– Neutrino energy (and cross sections)– Detector mass– Background level (e.g. hadronic component in cosmic rays)

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Reactor experiments• Search for disappearance of anti-νe

The ratio of observed to measured neutrino flux from reactor experiments as a function of their source distance L

SBL LBL VLBL

important to interpret Atmospheric neutrino data

important to interpret solar neutrino data

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KamLAND• Detect anti-νe produced by 53 reactors in Japan!

Schematics of the KamLAND detector: 1200 m3 of scintillator in a spherical balloon of 13 m

diameter and watched by 1879 PMTs

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Evidence for reactor anti-neutrino disappearance

Ratio of the measured to expected anti-neutrino spectrum versus L/E

• Deficit in the observed flux of electron anti-neutrinos (disappearance)R = 0.658 ± 0.044 ± 0.047

• The spectrum shows the signature of neutrino oscillations (L/E dependency)

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Accelerator neutrinos• Neutrinos produced by the decay of pions, kaons and

muons from a proton beam onto a target– Pion decay in flight: mostly muon neutrinos (OR anti-neutrinos)

with energies ~ GeV or more; e.g. SBL: CHORUS, NOMAD, CHARM, LSND; LBL: MINOS, OPERA, ICARUS, T2K …

– Muon decay at rest: muon anti-neutrinos of low energy from muon decay, with energy ~ tens MeV; e.g. KARMEN, LSND

– Beam dump: protons of very high energy are completely stopped by a target; muon and electron neutrinos with energy ~ 100 GeV

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LSND• All experiments did not find

any indication of oscillations

• Except LSND:– Signal in – Weak signal

• Combined analyses did not exclude this results

• Three-neutrinos mixing scheme to be extended (sterile neutrinos?)

• Design dedicated experiment: MiniBooNE

Region of squared-mass difference and mixing angle allowed at 90% CL by a combined analysis of LSND and KARMEN (green) and exclusion curves by KARMEN and other experiments

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MiniBooNE• Concept of sterile neutrino:

– non-interacting light particle– Singlet in the

SU(3)xSU(2)xU(1) group– It is mixed with active neutrinos

The MiniBooNE detector

• Test LSND studying • Changes:

– Higher energy (500 MeV compared to 30 MeV

– Longer baseline (500 m compared to 30 m)

The MiniBooNE excluded region compared with LSND results

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LBL accelerator experiments• K2K designed to test atmospheric

neutrino oscillations based on observation of muon neutrino disappearance. Beam: almost pure νµ

with mean energy 1.3 GeV• Other LBL: MINOS, ICARUS, OPERA

• K2K observed muon neutrino disappearance

Energy spectrum of the muon-neutrino events observed in the

K2K experiment

Best-fit with oscillations

No oscillations

important to interpret Atmospheric neutrino data

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Atmospheric neutrinos• Generated in the interaction of primary cosmic rays with the Earth’s

atmosphere

• Secondaries are generated which include all the hadrons and their decay products

• Energy spectrum is peaked at ~ GeV and extends to higher energies with a power-law

≈ 15 Km

Cosmic Ray

π+ µ+

e+

νeνµ

νµ

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The up-down symmetry/asymmetry1. The production of high energy atmospheric neutrinos is uniform

around the globe

2. A neutrino passing at point A with angle θ, reaches B at an angle π−θ

A

B

να

θ

π−θ

⇒The fluxes of neutrinos of a given flavor from opposite directions are the same at any location

⇒Up/down symmetry expected

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Atmospheric neutrinos• Mostly pions are produced, which decay into muons and

neutrinos

• In the 1960s (neutrino induced) muon tracks detected deep underground (~ 8000 mwe)

• The Kamiokande and IMB detectors: detect charged particles via Cherenkov radiation in water

• Observed less muon than expected• Atmospheric neutrino anomaly To detect charged particles, the

KAMIOKANDE detector utilizes Cherenkov radiation in the water.

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(Super-)kamiokande

The inside of the Superkamiokande detector

• Underground detector with arrays of PMTs:– IMB, (Super-)Kamiokande, SNO

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Wave front

Charged Particle v > c / n

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How to tell a νµ from a νe

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The up-down asymmetry• Asymmetry observed: a model independent proof of neutrino

oscillations

Up/down asymmetry of the neutrino flux as a function of the neutrino energy for the Kamioka and Sudan sites. Right: muon neutrinos. Left: electron neutrinos

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The atmospheric neutrino anomaly• First indication: the number of Sub-GeV muon-like events

was less than expected, while the number of electron-like events was compatible to the prediction

• Deficit of muon-like events

Zenith angle distribution of the through-going muon flux observed in Kamiokande

Data from Kamiokande alone cannot separate between νµ → νe and νµ → ντbut results from CHOOZ excluded νµ → νe

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Solution of the atm. anomaly: flavor• The results of CHOOZ and Paolo Verde disfavor νµ → νe

• The results of Superkamiokande favor νµ → ντ and disfavor νµ → νs (s=sterile neutrino)

• Confirmed by K2K (extremely important since rather different concepts and systematic uncertainties)

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Solar neutrinos• Powerful source of electron

neutrinos• Neutrino produced in two

groups of reactions:– pp chain– CNO cycle

• Energy ~ 1 MeV

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The pp chain of stellar thermonuclear reactions

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Standard Solar Model (SSM)• Rate of radio-chemical detectors is measured in Solar Neutrino Units (SNU) =

10-36 events atom-1s-1

Predicted energy spectra of neutrino fluxes

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Detection• Energy ~ MeV• σ = 10-44 cm2

• Interaction probability ~ 10-12

• Detection of solar neutrinos– First Homestake in 1970– Gallex/GNO in the ‘90s– Super-Kamiokande and SNO

later• Proof of the theory of

thermonuclear energy generation is stars!

• Discovery of the solar neutrino problem in favor of neutrino oscillations

Pauli: I have done a very bad thing today by proposing a particle that cannot be detected: it is something no theorist should ever do

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Detection of Solar Neutrinos• The Homestake experiment:

detect the radioactive Ar nucleus produced by interaction of a solar neutrino with the nucleus of a Cl atom (Eth=814 KeV):

• Expected 1.5 ± 0.6 atoms/day)• Found fewer (~ 1/3) neutrinos

than expected from the SSM ….

• Deficit confirmed by other experiments and at other energies

• The solar neutrino problemThe Homestake solar neutrino detector

(1,500 m underground to filter out cosmic particles, 615 ton of C2Cl4)

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Other radiochemical experiments

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GALLEX/SAGE results

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Electron scattering• Mostly electron neutrinos contribute to the process

• The cross section (T kinetic energy of the final electron)

• Strongly peaked for electron emission in the neutrino direction

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Solar neutrino anomaly

Angular distribution of solar neutrino event candidates of SuperKamiokande (SK, 50

kton water tank)

elastic scattering peak

background events

• Recoil electron have a sharp forward peak

• Flux measured ~ 1/2 of expected

22400 ± 230 solar neutrino events

SuperKamiokande image of the Sun

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Vacuum Oscillations of Solar ν’s• Pontecorvo & Gribov 1969• 1968, the Homestake experiment detects less neutrinos

than expected: the solar neutrino problem • Survival probability of solar neutrinos in case of two-ν’s

mixing (L distance Sun-Earth, L0=1 a.u., e eccentricity of Earth’s orbit):

• But from the analysis of solar neutrino data:– No significant seasonal variation observed– Energy spectra not compatible with the distortion expected due

to the transition probability • Vacuum oscillation are disfavored …

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Modulation of solar neutrinos• The only periodic variation in the rate of solar neutrinos agrees

with what expected due to the eccentricity of the Earth’s orbit.• No indication of other modulations due to neutrino oscillations

Seasonal variation of the solar ν flux measured in Superkamiokande

Solar neutrino flux as a function of time measured in Superkamiokande

Prediction based on the eccentricity of the Earth’s orbit

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The Sudbury Neutrino Observatory (SNO)• SNO detect solar neutrinos through

– CC: – NC:– ES:

• They provide a handle on– CC: energy spectrum of νe

– NC: total neutrino flux– ES: equivalent (independent) to SK,

measure the angular distribution of the events

• Measure Φ(νe) and ΣiΦ(νi)

• The flux of non-electron neutrinos (oscillated) is then: Φ(non−νe) = ΣiΦ(νi) - Φ(νe)

The SNO detector: one kiloton pure D2O in a spherical acrylic vessel

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Neutrino reactions in SNO

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Phys. Rev. C 75 045502 (2007)

Model prediction (no oscillations)

Flux of νµ and ντ as a function of νε

SNO results: solar νe deficit confirmed • The NC measurement of the total neutrino flux demonstrates that

about two electron neutrinos out of three change their flavor

• Fluxes needed to explain SNO data, assuming the energy spectrum of 8B:

• They largely disagree: proof that νe do change during propagation

• Good agreement between the NC SNO flux and what expected by the SSM

νe

Total ν flux

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Neutrino oscillations in vacuum• The flavor states να do not coincide with the mass

eigenstates νk

• The flavor states are combinations of the eigenstates νk:

• The mass states νk are eigenstates of the Hamiltonian:– Admixture of mass eigenstates in a given neutrino state do not

change (there is no ν1↔ ν2 transition)

– The phase difference between eigenstates increases monotonously

• The process is periodic. • The oscillation length is the distance at which the system

returns to its original state

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Resonant flavor transition in matter• Neutrinos in matter are subject to a potential due to

elastic scattering with the medium (electrons and nucleons), equivalent to an index of refraction

Feymann diagrams for the elastic scattering processes that generate the CC potential (VCC, left) and the NC potential (VNC, right). GF Fermi coupling

constant and Ne (Nn) number density of electrons (neutrons)

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Propagation in matter• In the presence of matter the Hamiltonian changes

Ho → H = Ho + V (Ho Hamiltonian in vacuum)

• The Schroedinger equation can be written in terms of matter mixing angle and effective squared-mass difference

• The eigenstates and the eigenvalues (and therefore the mixing angle) depend on the matter density and on the neutrino energy

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Matter effects in a medium with changing density

If the density changes during propagation:• The mixing angle changes

• The instantaneous eigenstates of the Hamitonian ν1m and ν2m are no longer eigenstates of propagation

• Transitions ν1m ↔ ν2m can take place

• If the density changes slowly (“adiabatic condition”) the transitions ν1m ↔ ν2m can be neglected

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Global fit of solar neutrino data• “Large mixing angle” solution of the solar neutrino

problem:– The mixing angle is large but not maximal

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Three neutrino mixing• The solar and atmospheric neutrino data provide evidence

of at least two squared-mass differences • Presence of sterile neutrinos disfavored in both cases• Three neutrino mixing: two independent squared mass

differences

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